HallsofIvy said:I just automatically looked at the discrimant for 3x2- 6x+ 2:
[tex]\sqrt{6^2- 4(3)(2)}= \sqrt{36- 24}= \sqrt{12}[/tex]
Since that is a real number, the 3x2- 6x+2 changes sign, and the original function changes "direction", at two places.
To prove the non-one-to-one property of a function, you need to show that there exist two distinct inputs that result in the same output. In other words, the function is not injective and therefore not one-to-one.
The function f(x)=x^3-3x^2+2x is a polynomial function with a degree of 3. It is a cubic function that can be graphed as a curve.
The interval (-k,k) represents a range of values for the input variable x. In this function, it means that the function is defined for all values of x between -k and k, including -k and k.
To prove that a function is not one-to-one, you can use a counterexample. This means finding two distinct inputs that result in the same output. In this case, we can find two values in the interval (-k,k) that give the same output for f(x)=x^3-3x^2+2x.
Proving the non-one-to-one property of a function is important because it tells us that the function is not invertible. This means that there is no unique inverse function for this function, and we cannot solve for x in terms of y. It also helps us understand the behavior of the function and its relationship between inputs and outputs.